91Ó°ÊÓ

A quadratic expression is a polynomial of degree two. It typically takes the form \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). Quadratic expressions are fundamental, appearing in various algebraic and real-world applications.
Here are some examples of quadratic expressions:

• \( x^2 + 5x + 6 \)
• \( 2x^2 - 7x + 3 \)
• \( 4x^2 - 2 \)

Quadratics can be solved using various methods such as factoring, completing the square, and using the quadratic formula.
In the current exercise, the expression \( r^{2}-6rs+9s^{2} \) is a quadratic in two variables \( r \) and \( s \). It’s structured as \( ar^2 + brs + cs^2 \), where \( a = 1, b = -6, \) and \( c = 9 \). Recognizing its quadratic form is essential for factoring.

A perfect square trinomial is a special type of quadratic expression that can be expressed as the square of a binomial. It follows the pattern \( (x + y)^2 = x^2 + 2xy + y^2 \) or \( (x - y)^2 = x^2 - 2xy + y^2 \).
Identifying a perfect square trinomial involves checking this pattern. For instance, the expression \( r^2 - 6rs + 9s^2 \) fits the pattern:

• \( (r)^2 = r^2 \)
• \(-2(r)(3s) = -6rs \)
• \((3s)^2 = 9s^2 \)

When these conditions match, the expression is a perfect square trinomial. This reveals that \( r^2 - 6rs + 9s^2 \) can be rewritten as \( (r - 3s)^2 \). Recognizing this allows for easier factoring into a binomial squared.

Trinomial factoring is the process of breaking down a quadratic trinomial into the product of two binomials. For simple cases like perfect square trinomials, this process can be straightforward.
Typically, trinomial factoring requires:

• Identifying patterns or structures within the expression
• Recognizing perfect squares and using related formulas
• Writing formulas in a simplified, factorable form

In the given exercise, recognizing \( r^2 - 6rs + 9s^2 \) as a perfect square trinomial simplifies the factoring greatly.
Instead of searching for two numbers that multiply to \( ac \) (product of the leading and constant terms) and sum to \( b \), we directly see the factorization as \( (r - 3s)^2 \).
This expedites finding the solution because it's part of a known form.